The present invention relates generally to magnetic resonance imaging (MRI), and more particularly to system and method of phase sensitive MRI reconstruction using partial k-space data to minimize data acquisition time (TE) while preserving phase information and reducing edge blurring in the reconstructed image.
In MR imaging, the scan time can be reduced by using a partial NEX, or alternatively, the echo time can be reduced by using a fractional echo. This moves the time to the echo peak closer to the start of the read-out gradient waveform than in a full echo. However, in all partial echo or half-Fourier reconstruction strategies, all phase information is lost. The present invention is a method and system for using the homodyne reconstruction algorithm to generate a complex-valued image from which phase information can be extracted.
In general, the synthesis of the missing k-space data assumes that the MR data is Hermitian for a real-valued image. That is: EQU F(-k.sub.x)=*(k.sub.x) [1]
where the * denotes a complex conjugate. If the k-space is divided into 4 quadrants, the data for at least two of the four quadrants is needed in order to generate an image. Therefore, either a partial echo (partially filled k.sub.x) or partial NEX (partially filled k.sub.y) can be used, but not both.
The following background is a review of the prior art homodyne method. If .function.(x) is the real-valued image and .phi.(r) is the spatially varying phase in the image, the expression for the complex valued image can be written as: EQU I(x)=.function.(x)exp(j.phi.(x))=.function..sub.L (x)exp(j.phi..sub.L (x))+.function..sub.H (x)exp(j.phi..sub.H (x)). [2]
This expression is a linear combination of the Fourier transforms of the low-pass and high-pass filtered k-space data, respectively. In homodyne reconstruction, the phase is assumed to be slowly varying and that .phi..sub.L (x).apprxeq..phi..sub.H (x). Therefore, if only one-half of the high-pass filtered data is available, this is equivalent to multiplying the high-pass filtered data by a Heaviside function such that the resulting image is given by: ##EQU1##
where {character pullout} denotes a convolution. Since the convolution term decays with 1/x and that the phase is slowly varying, Eqn. [3] can be rewritten as: ##EQU2##
If the available high frequency data is weighted by 2, Eqn. [4] can be written as: ##EQU3##
If the spatially varying phase term is divided out, the image is then the real-valued part of I.sub.H (x)exp(-j.phi..sub.L (x)), i.e.: EQU .function..sub.L (x)+.function..sub.H (x)=.function.(x)=Re(I.sub.H (x)exp(-j.phi..sub.L (x))), [6]
where the spatially varying phase is estimated from the phase of the Fourier transform of the low-pass filtered data. It is noted that in Eqn. [6], all phase information has now been lost. Note that Eqn. [6] could easily be written as .function.(x)=Re(I.sub.H (x))e.sup.-j.phi..sup..sub.L .sup.(x) where the phase in the image is the low spatial frequency phase. However, this phase is only an estimate and is of little use. Hence, this technique is not suitable for phase contrast reconstruction. Furthermore, the loss of phase information requires that the Fourier transform in the y direction be performed first, before the homodyne reconstruction is applied to the data in the x direction.
It would therefore be desirable to have a system and method capable of preserving magnitude and phase information in a partially acquired k-space data set that allows reduced data acquisition times and significantly improves edge blurring in the reconstructed MR image.